Underwater patch kits are the fix for small pool liner tears.
Yes, you can patch a small tear without draining the pool. That’s what underwater repair kits are made for.
You can find them at any pool supply store or online. They come with a patch and a special waterproof adhesive that cures under water. Clean the area as best you can (a scrub brush helps), cut the patch a little bigger than the tear, and apply the adhesive according to the directions. Then press it on and hold for a minute. Some kits include a weight to hold it in place while it sets.
The key is a clean surface. Algae or dirt will ruin the bond. Also, if the tear is near a seam or bigger than a few inches, you might need a more permanent fix. But for a simple slit or puncture? This works fine.
No need to replace the whole liner for a tiny rip.